We also considered alternative Bayesian monotone spline formulations based on restricting the sign of the coefficients [1]}, [2]} instead of squaring unconstrained functions. However, squaring unconstrained functions led to superior estimation performance in simulations, particularly when the non-negative function being estimated was close to 0. To see why, suppose \(g(x) = \lbrace \sum _{j=1}^{m} s_j(x) c_j\rbrace ^2\) , where \(\mathbf {s}(x) = (s_j(x))_{j=1}^{m}\)  denotes \(m\)  B-spline basis functions and \(\mathbf {c} = (c_1, \ldots , c_m)^{\text{T}}\)  denotes the vector of basis coefficients. If we assume \(\mathbf {c} \sim N(0, C)\)  for some covariance matrix \(C\) , then for each \(x\) , \(g(x) / \left[\mathbf {s}(x)^{\text{T}} \, C \, \mathbf {s}(x)\right] \sim \chi _1^2\) . Since the \(\chi _1^2\)  density function has an infinite spike at 0, the squared model provides superior shrinkage of the non-negative function towards 0 as compared to the model \(g(x) = \mathbf {s}(x)^{\text{T}} \mathbf {c}\) , where we assume the truncated Gaussian prior on \(\mathbf {c} \sim N(0, C) \mathbb {1}[c_1 \ge 0, \ldots , c_m \ge 0]\) .
